Based on the relationship between each polygon and the number inside it, the value of 'X' is ______. A triangle contains the number 6, a rectangle contains 24, a pentagon contains 120, and a hexagon contains X.
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Compare the number of sides of each polygon with the number written inside it.
Step 1: Look at how the number changes from one polygon to the next.
Going from the triangle to the rectangle, the number of sides goes from $3$ to $4$, and the value goes from $6$ to $24$. Check the ratio:
\[ \frac{24}{6}=4 \]
This matches the new side count, $4$. Step 2: Check the next step the same way.
Going from the rectangle to the pentagon, sides go from $4$ to $5$, and the value goes from $24$ to $120$. Check the ratio:
\[ \frac{120}{24}=5 \]
Again this matches the new side count, $5$. Step 3: State the rule found.
Each time the side count goes up by one, the value gets multiplied by that new side count. This is exactly how a factorial builds up, since $n!=(n-1)!\times n$. Step 4: Apply the rule to the hexagon.
The hexagon has $6$ sides, one more than the pentagon's $5$ sides. So multiply the pentagon's value by $6$:
\[ X=120\times6=720 \] Step 5: Conclude.
\[ \boxed{X=720} \]
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